Mathematica Bohemica, Vol. 142, No. 3, pp. 243-262, 2017
Existence of infinitely many weak solutions for some quasilinear $\vec{p}(x)$-elliptic Neumann problems
Ahmed Ahmed, Taghi Ahmedatt, Hassane Hjiaj, Abdelfattah Touzani
Received July 14, 2015. First published January 2, 2017.
Abstract: We consider the following quasilinear Neumann boundary-value problem of the type $ - \displaystyle\sum_{i=1}^N\frac{\partial}{\partial x_i}a_i\Big(x,\frac{\partial u}{\partial x_i}\Big) + b(x)|u|^{p_0(x)-2}u = f(x,u)+ g(x,u) &\text{in} \Omega, \quad\dfrac{\partial u}{\partial\gamma} = 0 &\text{on} \partial\Omega. $ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.
References: [1] G. Anello, G. Cordaro: Existence of solutions of the Neumann problem for a class of equations involving the $p$-Laplacian via a variational principle of Ricceri. Arch. Math. 79 (2002), 274-287. DOI 10.1007/s00013-002-8314-1 | MR 1944952 | Zbl 1091.35025 [2] E. Azroul, A. Barbara, M. B. Benboubker, H. Hjiaj: Entropy solutions for nonhomogeneous anisotropic $\Delta_{\vec p(\cdot)}$ problems. Appl. Math. 41 (2014), 149-163. DOI 10.4064/am41-2-3 | MR 3281367 | Zbl 1316.35107 [3] M. Bendahmane, M. Chrif, S. El Manouni: An approximation result in generalized anisotropic Sobolev spaces and applications. Z. Anal. Anwend. 30 (2011), 341-353. DOI 10.4171/ZAA/1438 | MR 2819499 | Zbl 1231.35065 [4] M.-M. Boureanu, V. D. Rădulescu: Anisotropic Neumann problems in Sobolev spaces with variable exponent. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 4471-4482. DOI 10.1016/j.na.2011.09.033 | MR 2927115 | Zbl 1262.35090 [5] R. Di Nardo, F. Feo, O. Guibé: Uniqueness result for nonlinear anisotropic elliptic equations. Adv. Differ. Equ. 18 (2013), 433-458. MR 3086461 | Zbl 1272.35092 [6] L. Diening, P. Harjulehto, P. Hästö, M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002 [7] X. Fan, C. Ji: Existence of infinitely many solutions for a Neumann problem involving the $p(x)$-Laplacian. J. Math. Anal. Appl. 334 (2007), 248-260. DOI 10.1016/j.jmaa.2006.12.055 | MR 2332553 | Zbl 1157.35040 [8] O. Guibé: Uniqueness of the renormalized solution to a class of nonlinear elliptic equations. On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments (A. Alvino et al., eds.). Quaderni di Matematica 23. Caserta (2008), 255-282. MR 2762168 | Zbl 1216.35036 [9] P. Harjulehto, P. Hästö: Sobolev inequalities with variable exponent attaining the values $1$ and $n$. Publ. Mat., Barc. 52 (2008), 347-363. DOI 10.5565/PUBLMAT_52208_05 | MR 2436729 | Zbl 1163.46022 [10] B. Kone, S. Ouaro, S. Traore: Weak solutions for anisotropic nonlinear elliptic equations with variable exponents. Electron. J. Differ. Equ. (electronic only) 2009 (2009), paper No. 144, 11 pages. MR 2565886 | Zbl 1182.35092 [11] O. Kováčik, J. Rákosník: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. MR 1134951 | Zbl 0784.46029 [12] M. Mihăilescu, G. Moroşanu: Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions. Appl. Anal. 89 (2010), 257-271. DOI 10.1080/00036810802713826 | MR 2598814 | Zbl 1187.35074 [13] B. Ricceri: A general variational principle and some of its applications. J. Comput. Appl. Math. 113 (2000), 401-410. DOI 10.1016/S0377-0427(99)00269-1 | MR 1735837 | Zbl 0946.49001 [14] M. Růžička: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748. Springer, Berlin (2000). DOI 10.1007/BFb0104029 | MR 1810360 | Zbl 0962.76001 [15] L. Zhao, P. Zhao, X. Xie: Existence and multiplicity of solutions for divergence type elliptic equations. Electron. J. Differ. Equ. (electronic only) 2011 (2011), paper No. 43, 9 pages. MR 2788662 | Zbl 1213.35227
Affiliations: Ahmed Ahmed, Taghi Ahmedatt, Abdelfattah Touzani, Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, BP 1796 Atlas, Fez, Morocco, e-mail: ahmedmath2001@gmail.com, taghi-med@hotmail.fr, atouzani07@gmail.com; Hassane Hjiaj, Department of Mathematics, Faculty of Sciences, Tetouan University Abdelmalek Essaadi, Quartier M'haneche II, Avenue Palestine, BP 2121, Tetouan 93000, Morocco, e-mail: hjiajhassane@yahoo.fr
